# Browsing by Subject "Tropical geometry"

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Item Compactified mirror families for positive log Calabi-Yau surfaces(2020-08-14) Lai, Jonathan En; Keel, Seán; Allcock, Daniel; Ben-Zvi, David; Hacking, PaulShow more This work develops a method to canonically compactify mirror families for positive pairs (Y,D), where Y is a projective rational surface and D is an anti-canonical cycle of rational curves. The compactified families can be identified with universal families associated to generalized marked pairs, a notion that is also developed here. In order to make this identification, period integrals for the mirror family are required. To carry this out, recent techniques from mirror symmetry and tropical geometry are used.Show more Item Realizability of tropical lines in the fan tropical plane(2013-08) Haque, Mohammad Moinul; Helm, David, doctor of mathematicsShow more In this thesis we construct an analogue in tropical geometry for a class of Schubert varieties from classical geometry. In particular, we look at the collection of tropical lines contained in the fan tropical plane. We call these tropical spaces "tropical Schubert prevarieties", and develop them after creating a tropical analogue for flag varieties that we call the "flag Dressian". Having constructed this tropical analogue of Schubert varieties we then determine that the 2-skeleton of these tropical Schubert prevarieties is realizable. In fact, as long as the lift of the fan tropical plane is in general position, only the 2-skeleton of the tropical Schubert prevariety is realizable.Show more Item Tropical Hurwitz spaces(2011-12) Katz, Brian Paul; Helm, David, doctor of mathematics; Keel, Sean; Katz, Eric; Allcock, Daniel; Ben-Zvi, David; Hassett, BrendanShow more Hurwitz numbers are a weighted count of degree d ramified covers of curves with specified ramification profiles at marked points on the codomain curve. Isomorphism classes of these covers can be included as a dense open set in a moduli space, called a Hurwitz space. The Hurwitz space has a forgetful morphism to the moduli space of marked, stable curves, and this morphism encodes the Hurwitz numbers. Mikhalkin has constructed a moduli space of tropical marked, stable curves, and this space is a tropical variety. In this paper, I construct a tropical analogue of the Hurwitz space in the sense that it is a connected, polyhedral complex with a morphism to the tropical moduli space of curves such that the degree of the morphism encodes the Hurwitz numbers.Show more Item Tropical matroid homology(2022-05-09) Alderete, Austin; Tran, Ngoc Mai; Schroeter, Benjamin; Payne, Samuel; Siebert, BerndShow more This thesis consists of independent projects on tropical matroid homology, graph recognition algorithms, and matroid classification. The first part provides an affirmative answer to a 2016 conjecture regarding matroid homology as well as sufficient conditions under which the results can be generalized. Accompanying this work is a polymake extension which allows for the computation of tropical matroid homology as well as the homology of a novel chain complex which resembles tropical matroid homology but is associated to the Chow ring of an arbitrary matroid. The second work provides an algorithm for determining whether a graph is 2-chordal bipartite, a class of graphs which have been connected to discrete statistical models, and we determine the complexity of this algorithm. As a natural extension, we introduce the class of k-chordal bipartite graphs and show that this class is empty for k ≥ 4.Show more